Let us now discuss the dimension \(n+1=2\) . It is known since 1886, see
N. Sonin [1]} and G. Darboux [2]}, that in this case every equation (REF )
is (equivalent to) the Euler-Lagrange equation of an nonautonomous Lagrangian. This result was improved in
[3]} where it was shown that for every reversible path structure \(\mathcal {P}\)  there exists a reversible Finsler metric whose geodesics are paths of the structure. The irreversible case is still open, see e.g. [4]} where the case when all paths are circles was investigated in details.
